Combination synchronization of fractional order n-chaotic systems using active backstepping design

Vijay K. Yadav 1  and S. Das 1
  • 1 Department of Mathematical Sciences, Indian Institute of Technology (BHU), 221005, Varanasi, India
Vijay K. Yadav
  • Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India
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and S. Das
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  • Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India
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Abstract

In this article, a scheme using active backstepping design method is proposed to achieve combination synchronization of n number of fractional order chaotic systems. In the proposed method the controllers are designed with the help of a new lemma and Lyapunov function in a systematic way. Synchronization among three/four fractional order systems have been shown as examples of synchronization of n-chaotic systems. Numerical simulation and graphical results clearly exhibit that the method of this new procedure is easy to implement and reliable for synchronization of fractional order chaotic systems.

1 Introduction

The theory of fractional calculus deals with derivatives and integrals of arbitrary order and has applications in various scientific field and engineering including viscoelasticity, fluid mechanics, material science, colored noise, dielectric polarization, electromagnetic wave, bioengineering, biological model, electromechanical system, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The fractional differential equations are the generalization of classical differential equations. The advantage of fractional order system is it allows greater flexibility in the model. The fractional order differential operator is non local but integer order differential operator is a local operator in the sense that fractional order differential operator takes into account the fact that the future state not only depends upon the present state but also upon all of the histories of its previous states. For this realistic property, fractional calculus which was in earlier stage considered as mathematical curiosity now becomes the object of extensive development of fractional order partial differential equations for its the purpose of engineering applications.

Chaos synchronization is an interesting phenomenon of nonlinear dynamical systems and it may occur when two or more chaotic systems are coupled or one chaotic system drives the other. The synchronization of chaotic systems was first given by Pecora and Carroll [12] in 1960, and after which it has been intensively studied due to its potential applications in various fields viz., ecological system, physical system, chemical system, secure communications etc [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In recent years various types of synchronization have been investigated such as complete synchronization, anti-synchronization, lag synchronization, adaptive synchronization, projective synchronization, function projective synchronization etc [23, 24, 25, 26] and also different schemes have been successfully applied to chaos synchronization viz., linear and nonlinear feedback control method, active control method, adaptive control method, sliding mode control method, backstepping method etc [27, 28, 29, 30, 31, 32, 33, 34].

The method here to use is backstepping design, which has been employed by many researchers for controlling and synchronizing chaotic systems as well as hyperchaotic systems. It consists in a recursive procedure that links the choice of a Lyapunov function with the design of a controller. Backstepping design is recognised as powerful design method for chaos synchronization. The design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems [35, 36, 37]. To stabilize and track chaotic systems, the method had been successfully used by Mascolo and Grassi [38] in the year 1999. In 2006, the backstepping control was used by Bin et al. [39] to synchronize two coupled chaotic neurons in external electrical stimulation. Wang and Ge [40] proposed the Adaptive synchronization of uncertain chaotic systems via backstepping design. Backstepping design was successfully applied by Tan et al. [41] during synchronization of the chaotic systems and also by Yu and Zhand [42] to control the uncertain behavior of chaotic systems. Recently, Park [43], Wu et al. [44] have shown that the backstepping method is very simple, reliable and powerful for controlling the chaotic behavior and synchronization of chaotic systems. But to the best of authors’ knowledge the synchronization of fractional order systems using backstepping control has not yet been studied by any researcher. The theme of the present study is to investigate the synchronization procedure for a number of fractional order chaotic systems using this simple and reliable backstepping method. In our earlier article [45], active control method and backstepping approach are used to synchronize the fractional order chaotic systems. The fractional order Chen and Qi systems are taken to synchronize using both the methods. It was shown that the backstepping method takes less time to synchronize as the systems pair approaches from standard order to fractional order. This has motivated the authors to find the required time of synchronization among three/four fractional order chaotic systems.

Initially the prediction of a system had been confined through finding the analytical solution of the formal modelling of the systems via mathematical modelling with a set of parameters and initial/boundary conditions. But after the advent of modern computers and related software packages, the simulation has become a useful technique of modelling of many streams of science and engineering as well as computational sociology. Nowadays it is used in technology to optimize the performance, safety engineering, also during modelling of natural and human systems. Simulation is described as the limitation of operation of a real world system over time. Thus before performing simulation, it requires to develop a model which will represent key features of the selected physical or abstract systems. Thus simulation basically represents the operation of the system over time. During synchronization of identical or non-identical chaotic systems the simulation is used to find requirement of minimum time after which the states of slave system behave similar to the master system.

The synchronization of three chaotic dynamical systems in integer order are first studied by the Runzi et al. [46] in 2011. In 2013 Zhang et al. [47] studied the combination synchronization of different kinds of spatiotemporal coupled systems with unknown parameters. In 2016 the combination synchronization of different kinds of spatiotemporal coupled systems with unknown parameters is studied by Wang et al. [48]. It is seen from literature survey that the synchronization between three and more chaotic systems are few in numbers. Runzi et al. [46] had stated the cause of investigation between two drive systems and one response system through a physical application in secure communication as transmitted signals can be split into several parts, each part loaded in different drive systems which shows that transmitted signals have stronger anti-attack ability and anti-translated capability than that transmitted by the usual transmission model. This has motivated the authors to study the generalization of synchronization between chaotic systems, when the systems have memory effects. The main contribution of the present scientific contribution is introduction of a new type of synchronization scheme known as combination synchronization which used to synchronize a number of fractional order chaotic systems. The backstepping method is applied during synchronization of fractional order chaotic systems using Lyapunov stability theory and a new lemma for Caputo derivative. The combination synchronization of three and four fractional order chaotic systems are found through numerical simulation which is presented graphically to show the effectiveness and feasibility of the proposed scheme and method. The salient feature of the present study is the pictorial presentation of requirement of less time during synchronization of four systems compared to three systems.

The article is organized as follows: In section 2, the definition and Lemma for Caputo derivative in fractional order case are introduced. Section 3 includes the combination synchronization scheme of fractional order n-chaotic systems are presented. In section 4, the fractional order chaotic systems are introduced. Sections 5 and 6 contain the combination synchronization of three and four chaotic systems using backstepping design respectively. Section 5.1 and 6.1 provides numerical simulation results and followed by a conclusion of the overall research work given in Section 7.

2 Basic definitions of Fractional order derivative and Lemma

The definitions of fractional order derivative given by B Riemann and J. Liouville and also by M. Caputo are as follows.

Definition 1

The Riemann–Liouville (R-L) fractional derivative operator of order q > 0 of a function f(t) is defined by [49]

Dxqft=dndtnJxnqft,n1<qn,nN,

where the fractional integral operator of order q > 0 of a function f(t) is given by

Jtqft=1Γq0ttξq1fξdξ,

with Jt0ft=ft..

Definition 2

The Caputo order fractional derivative of a function f(t) is [50, 51]

Dtqft=1Γnq0ttξnq1fnξdξ,n1<q<n,nN,
Dtqft=dnftdxn,q=n,

where Dtqf(t) satisfies the following basic property

JtqDtqft=ftk=0n1fk0+tkk!,t0,n1<q<n,nNandfCμn,μ1.

Between these two, the Caputo derivative is commonly used by the researchers. The main difference of considering the Caputo derivative is that its derivative of a constant is zero whereas in the case of finite value of lower terminal ′a′ the R-L derivative of a constant is aDtqc=ctqΓ(1q). However for the case a → −∞ this becomes zero. Thus for the steady state dynamical process fractional order derivative for both Caputo and R-L provide the same results.

Another reason of choosing Caputo derivative is that it holds for homogeneous and non-homogeneous initial conditions whereas R-L derivative require homogeneous initial condition during the solution of initial value problems.

Lemma 1

[52] Letx(t) ∈ R be a continuous and derivable function. Then for any time instant tt0,

12t0cDtqx2(t)x(t)t0cDtqx(t),q(0,1).

3 The scheme of combination synchronization of fractional order n-chaotic systems

In this scheme, (n-1) drive systems and one response system are assumed to be in fractional order system.

The drive systems are considered as

Dtqx1=f1(x1),
Dtqx2=f2(x2),
....Dtqxn1=fn1(xn1),

and the response system is taken as

Dtqxn=fn(xn)+U(x1,x2,......,xn),

where x1=(x11,x21,........,xn1),x2=(x12,x22,........,xn2),,xn1=(x1n1,x2n1,........,xnn1) and xn=(x1n,x2n,........,xnn) with x1, x2,   .........., xn − 1, xnRn are the state vectors of the n-chaotic systems. f1, f2, ......., fn − 1, fn : RnRn are the n-continuous vector functions and U(x1, x2, ......, xn) : Rn×Rn×.......×RnRnntimes is a controller which will be designed latter.

Definition

The fractional order (n-1) drive systems and one response system follow combination synchronization among n-chaotic systems if there exists n constant matrixes called scaling matrixes A1, A2, ....., AnRn with An ≠ 0 such that

limn+A1x1+A2x2+.......Anxn=0, where ∥ . ∥ represent the matrix norm.

It is noted that if A1 ≠ 0, A2 = A3 = ......... = An − 1 = 0, An = I then this problem is reduced to the projective synchronization, where I is an n × n identity matrix. If the scaling matrix A1 is considered as a function, then synchronization problem is reduced into function projective synchronization problem. Again if A1 = A2 = ......... = An − 1 = 0, then the problem becomes a chaos control problem.

4 Systems’ descriptions

4.1 Fractional order Newton-Leipnik system

The fractional order Newton-Leipnik system [53] was first studied in the year 2008, which is given by

dqx1dtq=a1x1+x2+10x2x3
dqx2dtq=x10.4x2+5x1x3
dqx3dtq=a2x35x1x2,0<q<1,

where a1 and a2 are the variable parameters’ and a2 ∈ (0, 0.8). The system is ill-behaved when a2 takes the values outside of this interval. If a2 becomes close to zero, the system shows uninteresting dynamic and if a2 ≥ 0.8, the given system becomes explosive i.e., the solution of this system diverges to infinity for any initial condition other than the critical points.

For the parameters’ values a1 = 0.4, a2 = 0.175 and the initial condition (0.19, 0, −0.18), the Newton-Leipnik system shows chaotic behaviour at q = 0.95 which is depicted through Fig. 1(a).

Fig. 1
Fig. 1

Phase portraits of fractional order (a) Newton-Leipnik system, (b) Liu system, (c) Lotka-Voltra system, (d) Chen system for the order of derivative q = 0.95.

Citation: Nonlinear Engineering 8, 1; 10.1515/nleng-2017-0073

4.2 Fractional order Liu system

The Liu system [54] was studied in the year 2009, which was later extended to fractional order Liu system by Gejji and Bhalekar [55] in 2010 as

dqy1dtq=b1y1b4y22
dqy2dtq=b2y2b5y1y3
dqy3dtq=b3y3+b6y1y2.

The phase portrait of the system is described through Fig. 1(b), which shows that the system exhibits chaos at the lowest fractional order q = 0.92 for the values of parameters b1 = 1, b2 = 2.5, b3 = 5, b4 = 1, b5 = 4, b6 = 4 and initial condition (0.2, 0, 0.5). The chaos control of Liu system is given in [56]. The trajectories of the system are controlled at its all equilibrium points for order of the derivative 0 < q ≤ 1.

4.3 Fractional order Lotka-Voltra system

The fractional order Lotka-Voltra system [57] is given as

dqz1dtq=c1z1c2z1z2+c5z12c6z3z12
dqz2dtq=c3z2+c4z1z2
dqz3dtq=c7z3+c6z3z12.

The chaotic attractor of the system is described through Fig. 1(c) at q = 0.95 for the values of the parameters c1 = c2 = c3 = c4 = 1, c5 = 2, c6 = 2.7, c7 = 3 and initial condition (1, 1.4, 1).

4.4 Fractional order Chen system

The fractional order Chen system [58] is considered as

dqw1dtq=d1(w2w1)
dqw2dtq=(d3d1)w1w1w3+d3w2
dqw3dtq=w1w2d2w3.

Fig. 1(d) shows the chaotic attractors of the system at the fractional order q = 0.95 for the parameters’ values d1 = 35, d2 = 3, d3 = 28 and the initial condition (1, 1.4, 1). The chaos control of Chen system is given in [58]. The system is controlled at order q = 0.9, q = 0.8 at its equilibrium points. The trajectories of the system is stable at all the equilibrium points of the system and it will also remain be controlled for order 0 < q ≤ 1.

5 Synchronization of fractional order Newton-Leipnik, Lotka-Voltra and Liu systems

For the study of synchronization among three fractional order chaotic systems, two systems Newton-Leipnik (5) and Lotka-Voltra (7) are considered as drive system-I and drive system-II and third system Liu system is considered as response system. The response system with the control functions u1, u2, u3 is defined as

dqy1dtq=b1y1b4y22+u1
dqy2dtq=b2y2b5y1y3+u2
dqy3dtq=b3y3+b6y1y2+u3.

Defining the error functions as ei = yizixi, i = 1, 2, 3, we obtain the error system as

dqe1dtq=b1e1b4e2+φ1+u1
dqe2dtq=b2e2b5e1e3b5e3(z1+x1)b5e1(z3+x3)+φ2+u2
dqe3dtq=b3e3+b6e1e2+b6e2(z1+x1)+b6e1(z2+x2)+φ3+u3,

where φ1 = −b1(z1 + x1) − b4(z2 + x2) − c1z1 + c2z1z2c5z12+c6z3z12+a1x1x210x2x3

φ2=b5(z1+x1)(z3+x3)+b2(z2+x2)+c3z2c4z1z2+x1+0.4x25x1x3φ3=b6(z1+x1)(z2+x2)b3(z3+x3)+c7z3c6z3z12a2x3+5x1x2.

Now the control functions would be designed using backstepping approach for combination synchronization of three fractional order chaotic systems.

Theorem 1

If the control functions are chosen as

u1=φ1,u2=b5v1(z3+x3)b2v2+b4v1v2φ2u3=b6v1(z2+x2)+(b5b6)v2(z1+x1)+(b5b6)v1v2φ3,

where v1 = e1, v2 = e2, v3 = e3, then the drive system I & II will be combination synchronized with response system.

Proof

To achieve the results, let us use the active backstepping procedure through following three steps.

Step-I: Defining v1 = e1, we get

dqv1dtq=dqe1dtq=b1v1b4e2+φ1+u1,

where e2 = α1(v1) is regarded as a virtual controller. For designing α1(v1) to stabilizev1 - subsystem, choosing the Lyapunov function V1 as

V1=12v12.

The-th order fractional derivative of V1 w. r. to t is

dqV1dtq=12dqv12dtqv1dqv1dtq(using Lemma-1)i.e.v1[b1v1b4α1(v1)+φ1+u1].

If we take α1(v1) = 0 and u1 = φ1, then dqV1dtqb1v12 < 0, which implies that subsystem (11) is asymptotically stable. Since virtual control function α1 (v1) is an estimate function, defining the error variable v2 between e2 and α1 (v1) as

v2=e2α2(v1),

we obtain the following (v1, v2) -subsystem as

dqv1dtq=b1v1b4v2dqv2dtq=b2v2b5v1e3b5e3(z1+x1)b5v1(z3+x3)+φ2+u2,

where e3 = α2 (v1, v2) is regarded as an virtual controller.

Step II: To stabilize (v1, v2) - subsystem (12), choose Lyapunov function as

V2=V1+12v22=12v12+12v22.

The order fractional derivative of V2 w. r. to t is

dqV2dtq=12dqv12dtq+12dqv22dtq

v1dqv1dtq+v2dqv2dtq, (from Lemma 1)

i.e. ≤ − b1v12b4v1v2 + v2[b2v2b5v1α2(v1, v2) − b5α2(v1, v2)(z1 + x1) − b5v1(z3 + x3) + φ2 + u2]

If α2(v1, v2) = 0 andu2 = b5v1(z3 + x3) − b2v2 + b4v1v2φ2, then dqV2dtqb1v12v22<0, which implies that (v1, v2)-subsystem (12) is asymptotically stable.

Again defining the error variable as

v3=e3α2(v1,v2),

the (v1, v2, v3) - subsystem becomes

dqv1dtq=b1v1b4v2dqv2dtq=v2+b4v1b5v1v3b5v3(z1+x1)dqv3dtq=b3v3+b6v1v2+b6v2(z1x1)+b6v1(z2+x2)+φ3+u3

Step III: To stabilize the (v1, v2, v3) - subsystem (13), choosing the Lyapunov function V3 as

V3=V2+12v32=12v12+12v22+12v32.

The fractional derivative of V3 is

dqV3dtq=12dqv12dtq+12dqv22dtq+12dqv32dtq

v1dqv1dtq+v2dqv2dtq+v3dqv3dtq, (from Lemma 1)

i.e. ≤ − b1v12v22b5v1v2v3b5v2v3 (z1 + x1) + v3[−b3v3 +b6v1v2 +b6v2 (z1 +x1) +b6v1 (z2 +x2) +φ3 + u3]

Taking u3 = −b6v1 (z2 +x2) +(b5b6) v2 (z1 +x1) +(b5b6) v1v2φ3, we obtain dqV3dtqb1v12v22b3v32<0. Thus the system is asymptotically stable. Thus for v1 = e1, v2 = e2α1(v1) = e2 and v3 = e3α2 (v1, v2) = e3, the error systems ei → 0, i = 1, 2, 3, which helps to obtain combination synchronization among the three considered fractional order systems.

5.1 Numerical simulation and results

In the numerical simulation the parameters’ values of the fractional order Newton-Leipnik, Lotka-Voltra and Liu systems are taken as a1 = 0.4, a2 = 0.175, and b1 = 1, b2 = 2.5, b3 = 5, b4 = 1, b5 = 4, b6 = 4 respectively. The initial conditions of two drive systems and response system are taken as (0.19, 0, −0.18), and (0.2, 0, 0.5) respectively. Fig. 2 shows the synchronization among three fractional order chaotic systems are achieved through backstepping approach at q = 0.95. Figs. 2(a), 2(b) and 2(c) depict the time response of the state trajectories xi(t) + zi(t) and yi(t), where i = 1, 2, 3 represent the drive systems (5), (7) and response system (9) respectively. The error states are displayed through Fig. 2(d).

Fig. 2
Fig. 2

Combination synchronization among three fractional order chaotic systems (5), (6) and (7) for fractional order q = 0.95: (a) between x1(t) +z1(t) and y1(t), (b) between x2(t) + z2(t) and y2(t), (c) between x3(t) + z3(t) and y3(t), (d) the evaluation of error function e1(t), e2(t) sand e3(t).

Citation: Nonlinear Engineering 8, 1; 10.1515/nleng-2017-0073

6 Synchronization of fractional order Newton-Leipnik, Liu, Lotka-Voltra systems and Chen system

In this section to synchronize four fractional order chaotic systems, we consider fractional order Newton-Leipnik system (5), fractional order Liu system (6) and fractional order Lotka-Voltra system (7) as the drive systems I, II and III respectively. The fractional order Chen system (8) is taken as response system with control function u1,u2,u3 as

dqw1dtq=d1(w2w1)+u1dqw2dtq=(d3d1)w1w1w3+d3w2+u2dqw3dtq=w1w2d2w3+u3.

Defining error functions as ei = wiziyixi, i = 1, 2, 3, we obtain the error system as

dqe1dtq=d1(e2e1)+ψ1+u1dqe2dtq=(d3d1)e1e1e3e3(z1+y1+x1)e1(z3+y3+x3)+d3e2+ψ2+u2dqe3dtq=e1e2+e2(z1+y1+x1)+e1(z2+y2+x2)d2e3+ψ3+u3,

where

ψ1=d1z2+(d1b4y2)y2+(d11)x2(d1+c1)z1(d1b1)y1(d1a1)x1+c2z1z2c5z12+c6z3z1210x2x3
ψ2=(d3d1)(z1+y1+x1)(z1+y1+x1)(z3+y3+x3)+d3(z2+y2+x2)+c3z2c4z1z2b2y2+b5y1y3+x1+0.4x25x1x3ψ3=(z1y1x1)(z2+y2+x2)d2(z3+y3+x3)+c7z3c6z3z12+b3y3b6y1y2a2x3+5x1x2

Next the control functions u1,u2 and u3 would be designed using backstepping approach for combination synchronization of four fractional order chaotic systems.

Theorem: 2

If the control functions are chosen as

u1=ψ1,u2=d3v2+v1(z3+y3+x3)d2v1v2ψ2,u3=v1(z2+y2+x2)ψ3

where v1 = e1, v2 = e2, v3 = e3, then the drive systems (5), (6) and (7) will be combination synchronized with response system (14).

Proof

For synchronization, backstepping procedure is used through following steps.

Step-I: Considering v1 = e1,

dqv1dtq=dqe1dtq=d1(e2e1)+ψ1+u1,

where e2 = α1 (v1) is regarded as a virtual controller. To stabilize v1 -subsystem, let us define the Lyapunov function V1 as

V1=12v12,

whose fractional derivative is

dqV1dtq=12dqv12dtqv1dqv1dtqi.e.,v1[d1(α1(v1)v1)+ψ1+u1]

Taking α1(v1) = 0 and u1 = − ψ1, we get dqV1dtqd1v12 < 0, which implies that v1-subsystem (16) is asymptotically stable. For the virtual control function α1(v1), a variable v2 between e2 and α1(v1) is defined as

v2=e2α1(v1).

Then (v1, v2)-subsystem is obtained as

dqv1dtq=d1(v2v1)dqv2dtq=(d3d1)v1v1e3e3(z1+y1+x1)v1(z3+y3+x3)+d3v2+ψ2u2

Let us consider v3 = α2 (v1, v2) is a virtual controller.

Step II: In this step to stabilize (v1, v2)-subsystem (17), let us define the Lyapunov function V2 as

V2=V1+12v22=12v12+12v22.

Now

dqV3dtq=12dqv12dtq+12dqv22dtqv1dqv1dtq+v2dqv2dtqi.e.,d1v1v2d1v12+v2[(d3d1)v1v1α2(v1,v2)α2(v1,v2)(z1+y1+x1)v1(z3+y3+x3)+d3v2+ψ2+u2]

Taking α2 (v1, v2) = 0 and u2 = −d3v2 + v1 (z3 +y3 +x3) − d2v1v2ψ2, we get dqV2dtqd1v12v22<0, which makes subsystem (17) asymptotically stable.

Considering v3 = e3α2 (v1, v2), the (v1, v2, v3) - subsystem is obtained as

dqv1dtq=d1(v2v1)dqv2dtq=v2d1v1v1v3v3(z1+y1+x1)dqv3dtq=v1v2+v2(z1+y1x1)+v1(z2+y2+x2)d2v3+ψ3+u3.

Step III: In order to stabilize (v1, v2, v3) - subsystem (18), choosing the Lyapunov function as

V3=V2+12v32=12v12+12v22+12v32,

we get

dqV3dtq=12dqv12dtq+12dqv22dtq+12dqv32dtqv1dqv1dtq+v2dqv2dtq+v3dqv3dtq,i.e.,v1[d1(v2v1)]+v2[v2d1v1v1v3v3(z1+y1+x1)]+v3[v1v2+v2(z1+y1+x1)+v1(z2+y2+x2)d2v3+ψ3+u3]=d1v12v22+v3[v1(z2+y2+x2)d2v3+ψ3+u3]

If u3=v1(z2+y2+x2)ψ3,dqV3dtqd1v12v22d2v32<0, negative definite. In view of v1 = e1, v2 = e2α1 (v1) = e2, v3 = e2α2 (v1, v2) = e3, the errors states ei → 0, i = 1, 2, 3 will converge to zero after a finite period of time, and thus the combination synchronization among four fractional order chaotic systems will be achieved.

6.1 Numerical simulation and results

During synchronization the earlier values of the parameters and initial conditions for drive systems I, II, III and response system are considered. The time step size is taken as 0.005. The synchronization among four fractional order chaotic systems are achieved through Fig. 3 using the same method at q = 0.95. Figs. 3(a), 3(b) and 3(c) show the time response of the states xi(t) + yi(t) +i(t) and wi(t), where i = 1, 2, 3 represent the drive systems (5), (6), (7) and the response system (8). The error states for this case are described through Fig. 3(d). It is noticed that it takes less time for synchronization among four systems (Fig. 3(d)) compared to that of three systems (Fig. 2(d)) for the considered systems in both fractional order as well as integer order case (Fig. 4).

Fig. 3
Fig. 3

Combination synchronization among four fractional order chaotic systems (5), (6), (7) and (8) for fractional order q = 0.95: (a) between x1(t) + y1(t) + z1 (t) and w1(t), (b) between x2 (t) + y2 (t) + z2 (t) and w2(t), (c) between x3 (t) + y3 (t) +z3(t) and w3(t), (d) the evaluation of error functions e1 (t), e2 (t) and e3(t).

Citation: Nonlinear Engineering 8, 1; 10.1515/nleng-2017-0073

Fig. 4
Fig. 4

The evaluation of error functions e1(t), e2(t) and e3(t) at q = 1: (a) for three systems; (b) for four systems.

Citation: Nonlinear Engineering 8, 1; 10.1515/nleng-2017-0073

7 Conclusion

In the present study, the combination synchronization among a number of fractional order drive and response systems is successfully demonstrated using backstepping method. For validation, the combination synchronization of three and four systems are considered separately taking two systems and three systems as drive system respectively, while one system as response system, which clearly exhibit that the applied method is effective and convenient to achieve global synchronization of a number of non-identical fractional order chaotic systems. The exhibition of requirement of less time during synchronization of four systems compared to three systems is one of the major contributions of the present study. It is worth mentioning that this scientific contribution of combination synchronization among the fractional order chaotic systems will be significant to the research community involved in the area of modelling of fractional order dynamical systems.

Acknowledgement

The authors are extending their heartfelt thanks to the revered reviewers for their valuable comments to upgrade the present manuscript.

References

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    Bagley, R.L., Calico, R.A., Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn., 1991, 14, 304–311.

    • Crossref
    • Export Citation
  • [2]

    Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech., 1984, 51, 199.

  • [3]

    Kulish, V.V., Lage, J.L., Application of fractional calculus to fluid mechanics. J. Fluids Eng., 2002, 124, 803–806.

    • Crossref
    • Export Citation
  • [4]

    Das, S., Tripathi, D., Pandey, S.K., Peristaltic flow of viscoelastic fluid with fractional maxwell model through a channel. Appl. Math. Comput., 2010, 215, 3645–3654.

  • [5]

    Carpinteri, A., Cornetti, P., Kolwankar, K.M., Calculation of the tensile and flexural strength of disordered materials using fractional calculus. Chaos Solitons & Fractals, 2004, 21, 623-632.

    • Crossref
    • Export Citation
  • [6]

    Sun, H.H., Abdelwahed, A.A., Onaral, B., Linear approximation for transfer function with a pole of fractional order. IEEE Trans. Automat. Control, 1984, 29, 441-444.

    • Crossref
    • Export Citation
  • [7]

    Heaviside, O., Electromagnetic Theory. New York, Chelsea, 1971.

  • [8]

    Magin, R.L., Fractional calculus in bioengineering, Part 3. Crit. Rev. Biomed. Eng., 2004, 32, 195–377.

    • Crossref
    • Export Citation
  • [9]

    Magin, R.L., Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl., 2010, 59, 1585–1593.

  • [10]

    Gokdogan, A., Merdan, M., Yildirim, A., A multistage differential transformation method for approximate solution of Hantavirus infection model. Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 1–8.

    • Crossref
    • Export Citation
  • [11]

    Sabatier, J., Poullain, S., Latteux, P., Thomas J.L., Oustaloup, A., Robust speed control of a low damped electromechanical system based on CRONE control: application to a four mass experimental test bench. Nonlinear Dyn., 2004, 38, 383–400.

    • Crossref
    • Export Citation
  • [12]

    Pecora, L.M., Carroll, T.L., Synchronization in chaotic systems, Phys. Rev. Lett., 1990, 64, 821-825.

    • Crossref
    • PubMed
    • Export Citation
  • [13]

    Blasius, B., Huppert A., Stone, L., Complex dynamics and phase synchronization in spatially extended ecological system. Nature, 1999, 399, 354–359.

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Lakshmanan M., Murali, K., Chaos in nonlinear oscillators: controlling and synchronization. Singapore: World Scientific, 1996.

  • [15]

    Han, S.K., Kerrer C., Kuramoto, Y., D-phasing and bursting in coupled neural oscillators. Phys Rev Lett., 1995, 75, 3190–3,

    • Crossref
    • PubMed
    • Export Citation
  • [16]

    Cuomo, K.M., Oppenheim, A.V., Circuit implementation of synchronized chaos with application to communication. Phys Rev Lett., 1993, 71, 65-8.

    • Crossref
    • Export Citation
  • [17]

    Murali, K., Lakshmanan, M.P., Secure communication using a compound signal using sampled-data feedback. Appl Math Mech., 2003, 11, 1309–15.

  • [18]

    Agrawal, S.K., Srivastava, M., Das, S., Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems. Nonlinear Dyn. 2012, 69, 2277–2288.

    • Crossref
    • Export Citation
  • [19]

    Chang, C.M., Chen, H.K., Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems. Nonlinear Dyn. 2010, 62, 851–858.

    • Crossref
    • Export Citation
  • [20]

    Cai, G., Hu, P., Li, Y., Modified function lag projective synchronization of a financial hyperchaotic system. Nonlinear Dyn. 2012, 69, 1457–1464.

    • Crossref
    • Export Citation
  • [21]

    Srivastava, M., Ansari, S.P., Agrawal, S.K., Das, S., Leung, A.Y.T., Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method. Nonlinear Dyn. 2014, 76, 905-914.

    • Crossref
    • Export Citation
  • [22]

    Luo C., Wang, X.Y., Chaos in the fractional-order complex Lorenz system and its synchronization, Nonlinear Dyn. 2013, 71, 241-257.

    • Crossref
    • Export Citation
  • [23]

    Wang, X.Y., Song, J.M., Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control, Commun Nonlinear Sci Numer Simulat. 2009, 14, 3351-3357.

    • Crossref
    • Export Citation
  • [24]

    Wang, X.Y., He, Y.J., Projective synchronization of fractional order chaotic system based on linear separation, Physics Letters A, 2008, 372, 435-441.

    • Crossref
    • Export Citation
  • [25]

    Das, S., Yadav, V.K., Stability Analysis, Chaos Control of Fractional Order Vallis and El-Nino Systems and Their Synchronization, IEEE/CAA Journal of Automatica Sinica, 2017, 4 (1), 114-124.

    • Crossref
    • Export Citation
  • [26]

    Wang, X.Y., Zhang, X., Ma, C., Modified projective synchronization of fractional-order chaotic systems via active sliding mode control, Nonlinear Dyn. 2012, 69, 511–517.

    • Crossref
    • Export Citation
  • [27]

    Chen S.H., Lu, J., Parameters identification and synchronization of chaotic systems based upon adaptive control. Phys. Lett. A, 2002, 299, 353–8.

    • Crossref
    • Export Citation
  • [28]

    Huang, L., Feng, R., Wang, M., Synchronization of chaotic systems via nonlinear control. Phys. Lett. A, 2004, 320, 271–5.

    • Crossref
    • Export Citation
  • [29]

    Agrawal, S.K., Srivastava, M., Das, S., Synchronization of fractional order chaotic systems using active control method. Chaos Solitons & Fractals, , 2012.

    • Crossref
    • Export Citation
  • [30]

    Chen, S.H., Lu, J.: Synchronization of an uncertain unified chaotic system via adaptive control. Chaos Solitons & Fractals, 2002, 14, 643-7.

    • Crossref
    • Export Citation
  • [31]

    Razminia, A., Baleanu, D., Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics, 2013, 23, 873-879.

    • Crossref
    • Export Citation
  • [32]

    Park, J.H., Synchronization of Genesio chaotic system via backstepping approach. Chaos & Solitons and Fractals, 2006, 27, 1369-1375.

    • Crossref
    • Export Citation
  • [33]

    Lin, D., Wang, X., Nian, F., Zhang Y., Dynamic fuzzy neural networks modeling and adaptive backstepping tracking control of uncertain chaotic systems, Neurocomputing, 2010, 73, 2873–2881.

    • Crossref
    • Export Citation
  • [34]

    Tang, Q., Wang, X., Backstepping generalized synchronization for neural network with delays based on tracing control method, Neural Comput & Applic, 2014, 24, 775–778.

    • Crossref
    • Export Citation
  • [35]

    Zhang, H., Ma, X., Li, M., Zou, J., Controlling and tracking hyperchaotic Rossler system via active backstepping design. Chaos Solitons and Fractals, 2005, 26, 353-361.

    • Crossref
    • Export Citation
  • [36]

    Kokotovic, P.V., The joy of feedback: nonlinear and adaptive. IEEE Control Syst Mag, 1992, 6, 7-17.

  • [37]

    Krstic, M., Kanellakopoulus, I., Kokotovic, P., Nonlinear and adaptive control design John Wiley, New York, 1995.

  • [38]

    Mascolo, S., Grassi, G., Controlling chaotic dynamics using backstepping design with application to the Lorenz system and Chua’s circuit. Int J Bifur Chaos, 1999, 9, 1425–1434.

    • Crossref
    • Export Citation
  • [39]

    Bin, D., Jiang, W., Xiang yang, F., Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control, Chaos Solitons & Fractals, 2006, 29, 182-189.

    • Crossref
    • Export Citation
  • [40]

    Wang, C., Ge, S.S., Adaptive synchronization of uncertain chaotic systems via backstepping design, Chaos Solitons & Fractals, 2001, 12, 1199-206.

    • Crossref
    • Export Citation
  • [41]

    Tan, X.H., Zhang, J.Y., Yang, Y.R., Synchronization chaotic systems using backstepping design, Chaos Solitons & Fractals, 2003, 16, 37-45.

    • Crossref
    • Export Citation
  • [42]

    Yu, Y.G., Zhang, S.C., Controlling uncertain system using backstepping design, Chaos Solitons & Fractals, 2003, 15, 897–902.

    • Crossref
    • Export Citation
  • [43]

    Park, J.H., Synchronization of Genesio chaotic system via backstepping approach, Chaos Solitons & Fractals, 2006, 27, 1369-1375.

    • Crossref
    • Export Citation
  • [44]

    Wu, Y., Zhou, X., Chen, J., Hui, B., Chaos synchronization of a new 3D chaotic system, Chaos Solitons & Fractals, 2009, 42, 1812-1819.

    • Crossref
    • Export Citation
  • [45]

    Singh, A.K. Yadav, V.K., Das, S., Comparative study of synchronization methods of fractional order chaotic systems, Nonlinear Engineering, 2016, 5(3), 185–192.

  • [46]

    Runzi, L., Yinglan, W., Shucheng, D., Combination synchronization of three classic chaotic systems using active backstepping design. Chaos, 2011, 21, 043114.

    • Crossref
    • PubMed
    • Export Citation
  • [47]

    Zhang H., Wang X., Lin X., Combination synchronization of different kinds of spatiotemporal coupled systems with unknown parameters, IET Control Theory and Applications, 2013,

    • Crossref
    • Export Citation
  • [48]

    Wang, S., Wang, X., Wang, X., Zhou, Y., Adaptive generalized combination complex synchronization of uncertain real and complex nonlinear systems, AIP ADVANCES, 2016, 6, 045011.

    • Crossref
    • Export Citation
  • [49]

    Podlubny, I. Fractional Differential Equations, Academic Press, San Diego. CA, 1999.

  • [50]

    Gorenflo, R., Mainradi, F., Essentials of fractional calculus. Preprint submitted to Maphysto centre, Preliminary version 2000.

  • [51]

    Oldham, K., Spanier, J., The fractional calculus, Academic Press. New York- London 1974.

  • [52]

    Norelys, A.C., Manuel, A.D.M., Gallegos, J.A., Lyapunov functions for fractional order systems. Commun Nonlinear Sci. Numer. Simulat. 2014, 19, 2951–2957.

    • Crossref
    • Export Citation
  • [53]

    Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M., Chen, W.C., Lin, K.T., Kang, Y., Chaos in the Newton-Leipnik system with fractional order. Chaos, Solitons and Fractals, 2008, 36, 98–103.

    • Crossref
    • Export Citation
  • [54]

    Liu, C., Liu, L., Liu, T., A novel three-dimensional autonomous chaos system. Chaos, Solitons and Fractals, 2009, 39, 1950-1958.

    • Crossref
    • Export Citation
  • [55]

    Gejji, V.D., Bhalekar, S., Chaos in fractional ordered Liu system. Computers and Mathematics with Applications, 2010, 59, 1117-1127.

    • Crossref
    • Export Citation
  • [56]

    Wang, X.Y., Wang, M.J., Dynamic analysis of the fractional-order Liu system and its synchronization, CHAOS, 2007, 17, 033106.

    • Crossref
    • PubMed
    • Export Citation
  • [57]

    Petras, I., Fractional order nonlinear systems, modelling, analysis and simulation. Beijing, Berlin, Heidelberg: Higher education press, Springer-Verlag, 2011.

  • [58]

    Li, C., Chen, G., Chaos in the fractional order Chen system and its control. Chaos, Solitons and Fractals, 2004, 22, 549-554.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Bagley, R.L., Calico, R.A., Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn., 1991, 14, 304–311.

    • Crossref
    • Export Citation
  • [2]

    Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity. J. Appl. Mech., 1984, 51, 199.

  • [3]

    Kulish, V.V., Lage, J.L., Application of fractional calculus to fluid mechanics. J. Fluids Eng., 2002, 124, 803–806.

    • Crossref
    • Export Citation
  • [4]

    Das, S., Tripathi, D., Pandey, S.K., Peristaltic flow of viscoelastic fluid with fractional maxwell model through a channel. Appl. Math. Comput., 2010, 215, 3645–3654.

  • [5]

    Carpinteri, A., Cornetti, P., Kolwankar, K.M., Calculation of the tensile and flexural strength of disordered materials using fractional calculus. Chaos Solitons & Fractals, 2004, 21, 623-632.

    • Crossref
    • Export Citation
  • [6]

    Sun, H.H., Abdelwahed, A.A., Onaral, B., Linear approximation for transfer function with a pole of fractional order. IEEE Trans. Automat. Control, 1984, 29, 441-444.

    • Crossref
    • Export Citation
  • [7]

    Heaviside, O., Electromagnetic Theory. New York, Chelsea, 1971.

  • [8]

    Magin, R.L., Fractional calculus in bioengineering, Part 3. Crit. Rev. Biomed. Eng., 2004, 32, 195–377.

    • Crossref
    • Export Citation
  • [9]

    Magin, R.L., Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl., 2010, 59, 1585–1593.

  • [10]

    Gokdogan, A., Merdan, M., Yildirim, A., A multistage differential transformation method for approximate solution of Hantavirus infection model. Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 1–8.

    • Crossref
    • Export Citation
  • [11]

    Sabatier, J., Poullain, S., Latteux, P., Thomas J.L., Oustaloup, A., Robust speed control of a low damped electromechanical system based on CRONE control: application to a four mass experimental test bench. Nonlinear Dyn., 2004, 38, 383–400.

    • Crossref
    • Export Citation
  • [12]

    Pecora, L.M., Carroll, T.L., Synchronization in chaotic systems, Phys. Rev. Lett., 1990, 64, 821-825.

    • Crossref
    • PubMed
    • Export Citation
  • [13]

    Blasius, B., Huppert A., Stone, L., Complex dynamics and phase synchronization in spatially extended ecological system. Nature, 1999, 399, 354–359.

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Lakshmanan M., Murali, K., Chaos in nonlinear oscillators: controlling and synchronization. Singapore: World Scientific, 1996.

  • [15]

    Han, S.K., Kerrer C., Kuramoto, Y., D-phasing and bursting in coupled neural oscillators. Phys Rev Lett., 1995, 75, 3190–3,

    • Crossref
    • PubMed
    • Export Citation
  • [16]

    Cuomo, K.M., Oppenheim, A.V., Circuit implementation of synchronized chaos with application to communication. Phys Rev Lett., 1993, 71, 65-8.

    • Crossref
    • Export Citation
  • [17]

    Murali, K., Lakshmanan, M.P., Secure communication using a compound signal using sampled-data feedback. Appl Math Mech., 2003, 11, 1309–15.

  • [18]

    Agrawal, S.K., Srivastava, M., Das, S., Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems. Nonlinear Dyn. 2012, 69, 2277–2288.

    • Crossref
    • Export Citation
  • [19]

    Chang, C.M., Chen, H.K., Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems. Nonlinear Dyn. 2010, 62, 851–858.

    • Crossref
    • Export Citation
  • [20]

    Cai, G., Hu, P., Li, Y., Modified function lag projective synchronization of a financial hyperchaotic system. Nonlinear Dyn. 2012, 69, 1457–1464.

    • Crossref
    • Export Citation
  • [21]

    Srivastava, M., Ansari, S.P., Agrawal, S.K., Das, S., Leung, A.Y.T., Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method. Nonlinear Dyn. 2014, 76, 905-914.

    • Crossref
    • Export Citation
  • [22]

    Luo C., Wang, X.Y., Chaos in the fractional-order complex Lorenz system and its synchronization, Nonlinear Dyn. 2013, 71, 241-257.

    • Crossref
    • Export Citation
  • [23]

    Wang, X.Y., Song, J.M., Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control, Commun Nonlinear Sci Numer Simulat. 2009, 14, 3351-3357.

    • Crossref
    • Export Citation
  • [24]

    Wang, X.Y., He, Y.J., Projective synchronization of fractional order chaotic system based on linear separation, Physics Letters A, 2008, 372, 435-441.

    • Crossref
    • Export Citation
  • [25]

    Das, S., Yadav, V.K., Stability Analysis, Chaos Control of Fractional Order Vallis and El-Nino Systems and Their Synchronization, IEEE/CAA Journal of Automatica Sinica, 2017, 4 (1), 114-124.

    • Crossref
    • Export Citation
  • [26]

    Wang, X.Y., Zhang, X., Ma, C., Modified projective synchronization of fractional-order chaotic systems via active sliding mode control, Nonlinear Dyn. 2012, 69, 511–517.

    • Crossref
    • Export Citation
  • [27]

    Chen S.H., Lu, J., Parameters identification and synchronization of chaotic systems based upon adaptive control. Phys. Lett. A, 2002, 299, 353–8.

    • Crossref
    • Export Citation
  • [28]

    Huang, L., Feng, R., Wang, M., Synchronization of chaotic systems via nonlinear control. Phys. Lett. A, 2004, 320, 271–5.

    • Crossref
    • Export Citation
  • [29]

    Agrawal, S.K., Srivastava, M., Das, S., Synchronization of fractional order chaotic systems using active control method. Chaos Solitons & Fractals, , 2012.

    • Crossref
    • Export Citation
  • [30]

    Chen, S.H., Lu, J.: Synchronization of an uncertain unified chaotic system via adaptive control. Chaos Solitons & Fractals, 2002, 14, 643-7.

    • Crossref
    • Export Citation
  • [31]

    Razminia, A., Baleanu, D., Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics, 2013, 23, 873-879.

    • Crossref
    • Export Citation
  • [32]

    Park, J.H., Synchronization of Genesio chaotic system via backstepping approach. Chaos & Solitons and Fractals, 2006, 27, 1369-1375.

    • Crossref
    • Export Citation
  • [33]

    Lin, D., Wang, X., Nian, F., Zhang Y., Dynamic fuzzy neural networks modeling and adaptive backstepping tracking control of uncertain chaotic systems, Neurocomputing, 2010, 73, 2873–2881.

    • Crossref
    • Export Citation
  • [34]

    Tang, Q., Wang, X., Backstepping generalized synchronization for neural network with delays based on tracing control method, Neural Comput & Applic, 2014, 24, 775–778.

    • Crossref
    • Export Citation
  • [35]

    Zhang, H., Ma, X., Li, M., Zou, J., Controlling and tracking hyperchaotic Rossler system via active backstepping design. Chaos Solitons and Fractals, 2005, 26, 353-361.

    • Crossref
    • Export Citation
  • [36]

    Kokotovic, P.V., The joy of feedback: nonlinear and adaptive. IEEE Control Syst Mag, 1992, 6, 7-17.

  • [37]

    Krstic, M., Kanellakopoulus, I., Kokotovic, P., Nonlinear and adaptive control design John Wiley, New York, 1995.

  • [38]

    Mascolo, S., Grassi, G., Controlling chaotic dynamics using backstepping design with application to the Lorenz system and Chua’s circuit. Int J Bifur Chaos, 1999, 9, 1425–1434.

    • Crossref
    • Export Citation
  • [39]

    Bin, D., Jiang, W., Xiang yang, F., Synchronizing two coupled chaotic neurons in external electrical stimulation using backstepping control, Chaos Solitons & Fractals, 2006, 29, 182-189.

    • Crossref
    • Export Citation
  • [40]

    Wang, C., Ge, S.S., Adaptive synchronization of uncertain chaotic systems via backstepping design, Chaos Solitons & Fractals, 2001, 12, 1199-206.

    • Crossref
    • Export Citation
  • [41]

    Tan, X.H., Zhang, J.Y., Yang, Y.R., Synchronization chaotic systems using backstepping design, Chaos Solitons & Fractals, 2003, 16, 37-45.

    • Crossref
    • Export Citation
  • [42]

    Yu, Y.G., Zhang, S.C., Controlling uncertain system using backstepping design, Chaos Solitons & Fractals, 2003, 15, 897–902.

    • Crossref
    • Export Citation
  • [43]

    Park, J.H., Synchronization of Genesio chaotic system via backstepping approach, Chaos Solitons & Fractals, 2006, 27, 1369-1375.

    • Crossref
    • Export Citation
  • [44]

    Wu, Y., Zhou, X., Chen, J., Hui, B., Chaos synchronization of a new 3D chaotic system, Chaos Solitons & Fractals, 2009, 42, 1812-1819.

    • Crossref
    • Export Citation
  • [45]

    Singh, A.K. Yadav, V.K., Das, S., Comparative study of synchronization methods of fractional order chaotic systems, Nonlinear Engineering, 2016, 5(3), 185–192.

  • [46]

    Runzi, L., Yinglan, W., Shucheng, D., Combination synchronization of three classic chaotic systems using active backstepping design. Chaos, 2011, 21, 043114.

    • Crossref
    • PubMed
    • Export Citation
  • [47]

    Zhang H., Wang X., Lin X., Combination synchronization of different kinds of spatiotemporal coupled systems with unknown parameters, IET Control Theory and Applications, 2013,

    • Crossref
    • Export Citation
  • [48]

    Wang, S., Wang, X., Wang, X., Zhou, Y., Adaptive generalized combination complex synchronization of uncertain real and complex nonlinear systems, AIP ADVANCES, 2016, 6, 045011.

    • Crossref
    • Export Citation
  • [49]

    Podlubny, I. Fractional Differential Equations, Academic Press, San Diego. CA, 1999.

  • [50]

    Gorenflo, R., Mainradi, F., Essentials of fractional calculus. Preprint submitted to Maphysto centre, Preliminary version 2000.

  • [51]

    Oldham, K., Spanier, J., The fractional calculus, Academic Press. New York- London 1974.

  • [52]

    Norelys, A.C., Manuel, A.D.M., Gallegos, J.A., Lyapunov functions for fractional order systems. Commun Nonlinear Sci. Numer. Simulat. 2014, 19, 2951–2957.

    • Crossref
    • Export Citation
  • [53]

    Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M., Chen, W.C., Lin, K.T., Kang, Y., Chaos in the Newton-Leipnik system with fractional order. Chaos, Solitons and Fractals, 2008, 36, 98–103.

    • Crossref
    • Export Citation
  • [54]

    Liu, C., Liu, L., Liu, T., A novel three-dimensional autonomous chaos system. Chaos, Solitons and Fractals, 2009, 39, 1950-1958.

    • Crossref
    • Export Citation
  • [55]

    Gejji, V.D., Bhalekar, S., Chaos in fractional ordered Liu system. Computers and Mathematics with Applications, 2010, 59, 1117-1127.

    • Crossref
    • Export Citation
  • [56]

    Wang, X.Y., Wang, M.J., Dynamic analysis of the fractional-order Liu system and its synchronization, CHAOS, 2007, 17, 033106.

    • Crossref
    • PubMed
    • Export Citation
  • [57]

    Petras, I., Fractional order nonlinear systems, modelling, analysis and simulation. Beijing, Berlin, Heidelberg: Higher education press, Springer-Verlag, 2011.

  • [58]

    Li, C., Chen, G., Chaos in the fractional order Chen system and its control. Chaos, Solitons and Fractals, 2004, 22, 549-554.

    • Crossref
    • Export Citation
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The purpose of the Journal of Nonlinear Engineering is to provide a medium for dissemination of original research results in theoretical, experimental, practical, and applied nonlinear phenomena in engineering. The journal serves as a forum to exchange new ideas and applications of nonlinear problems occurring in aeronautical, biological, civil, chemical, communication, electrical, industrial, mechanical, mathematical, physical, and structural systems.

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  • View in gallery

    Phase portraits of fractional order (a) Newton-Leipnik system, (b) Liu system, (c) Lotka-Voltra system, (d) Chen system for the order of derivative q = 0.95.

  • View in gallery

    Combination synchronization among three fractional order chaotic systems (5), (6) and (7) for fractional order q = 0.95: (a) between x1(t) +z1(t) and y1(t), (b) between x2(t) + z2(t) and y2(t), (c) between x3(t) + z3(t) and y3(t), (d) the evaluation of error function e1(t), e2(t) sand e3(t).

  • View in gallery

    Combination synchronization among four fractional order chaotic systems (5), (6), (7) and (8) for fractional order q = 0.95: (a) between x1(t) + y1(t) + z1 (t) and w1(t), (b) between x2 (t) + y2 (t) + z2 (t) and w2(t), (c) between x3 (t) + y3 (t) +z3(t) and w3(t), (d) the evaluation of error functions e1 (t), e2 (t) and e3(t).

  • View in gallery

    The evaluation of error functions e1(t), e2(t) and e3(t) at q = 1: (a) for three systems; (b) for four systems.